november 2010 volume 1 number 2 - Advances in Electronics and ...

52 ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 2, NOVEMBER 2010
result of all tests from the NIST test suite, we must use
many source generators, which is numerically inefficient. In
this section, we introduce a new method for generating many
source streams by a single MCPG. The generator is derived
from the sawtooth chaotic map implemented in a finite-state
machine in the modular arithmetic. The benefit is that we can
combine many source streams into a single sequence without
significantly decreasing the speed of producing pseudorandom
numbers.
Let Sλ denotethesawtoothmap,namedalsotheRényimap,
the Bernoulli shift, or the Bernoulli map. Map Sλ transforms
the unit interval I = [0, 1) ⊂ X, X ≡ R into itself and has
the following form
Sλ(x) = λx mod 1, (2)
where λ is a real number. Computing successive values of
expression
sn = ⌊αxn⌋ , α ≥ 2 , n = 1, 2, ..., (3)
where α is an integer and xn = λxn−1 mod 1, we obtain
a sequence {sn} of integer numbers. Numbers sn can be
regardedas indices of subintervalscontaining xn and obtained
as the result of partitioning I into α disjoint, equal-sized
subintervals Ij, j = 0, 1, 2, ..., α − 1, covering the whole
set I. Through assigning a unique number (symbol) from
set Aα = {0, 1, ..., α − 1} to every Ij, the macroscopic
behavior of the dynamical system (Sλ, I) can be studied.
This macroscopic dynamics is called symbolic dynamics. It is
knownthatsymbolicsequencesmaybetreatedastrulyrandom
sequences in many aspects [7]–[10]. Assuming integer λ and
rational x0 = (p0)/(q0), where 0 < pn < q0, we obtain that
[11] ⎧ ⎨
⎩
sn = ⌊αxn⌋
xn = pn
q0
pn = λ · pn−1 mod q0
n = 1, 2, . . .
. (4)
Because in a finite-state machine the number of bits encoding
the values of all variables is limited to l, where l is finite,
expression (4) can be written as
⎧
⎪⎨ sn = ⌊α · xn⌋ � �
pn
xn = truncl n = 1, 2, . . . , (5)
q0
⎪⎩
pn = λpn−1 mod q0
where truncl denotes the truncation operation, leaving l the
most significant bits of quotient (pn)/(q0). If α = 2k , 1 ≤
k ≤ l, then sequence {sn} consists of numbers encoded by
the k most significant bits of xn. If additionally q0 = 2l or
q0 = 2l − 1, these bits are the same as the most significant
bits of pn (see [11] for examples). Then (5) is reduced to
�
sn = trunck(pn)
. (6)
pn = λpn−1 mod q0
The second formula in (6) describes the multiplicative congruential
pseudorandom generator (1) with a = λ and b = q0.
For α = 2 k , 1 ≤ k ≤ l and q0 = 2 l or q0 = 2 l − 1,
sequence {sn} is the same as the output sequence of the
truncated multiplicative congruentialpseudorandomgenerator.
To improvethe statistical propertiesof {pn}, successive pn are
first written into Table T with L cells, addressed from 0 to
L − 1. Next, we read off K numbers T1, T2, ..., TK from T
per one iteration of equation (6), where it is assumed that
L ≥ αK. The addresses of T1, T2, ..., TK depend on sn.
Numbers T1, T2, ..., TK are treated as vectors encoded by l
bits. The elements of K vectors are summed modulo 2 and
added modulo 2 to current number pn, denoted for clarity as
T0, forming a single vector Un. Its elements can encode an
integer number from interval (0, 2 l ) or a real number from
unit interval I = (0, 1). The pseudocode of an algorithm
proposed for producing {Un} has the following form:
Algorithm 1 Algorithm CPRNG
Initialization:
Choose k, p0 ∈ (0, q0) and the size L of Table T;
Write p0 into the first cell of Table T, i.e. T [0] := p0;
for n := � 1 to L − 1 do
pn := λpn−1 mod q0, n = 1, 2, ...L − 1
(7)
T [n] := pn
end for
Computations:
for n := 1 to N do
⎧
pn+L−1 := λpn+L−2 mod q0
⎪⎨
j := n mod L, L ≥ αK, α = 2
⎪⎩
k , 1 ≤ k ≤ l
T [j] := pn+L−1
s ′ n+L−1 := 1 + trunck(pn+L−1)
Un := T [j] ⊕ T [ � j + s ′ �
n+L−1 mod L]
⊕ · · · ⊕ T [ � j + Ks ′ (8)
�
n+L−1 mod L]
end for
In (8) it is that s ′ n+L−1 = 1+sn+L−1. The combined pseudorandom
number generator CPRNG repeatedly uses the “bit
stripping”,known from the shufflingalgorithmsof Gebhardor
BaysandDurham(seep.10in[2]).Numbers pn writteninto T
can be regarded as digits encoding a certain number p, written
in the fixed-point number system with base q0. If {pn} is a
random sequence, then all sequences composed of digits chosen
from digits encoding p are independent [2]. The addresses
of numbers T0, T1, .., TK differ by a constant value s ′ n+L−1 .
Numbers s ′ n+L−1 are the elements of symbolicsequence {sn}
produced by chaotic Sλ and realized in computer in the
modulararithmetic – shifted by unity. The same algorithm can
be used for other values q0 but symbols s ′ n+L−1 have to be
computed from formula s ′ n+L−1 = 1 + trunck(pn+L−1/q0),
i.e., they cannot be the most significant digits of pn+L−1
increased by 1. Changing the method of addressing Table T,
we can obtain different combined generators.
The period mu ofsequence {Un} dependson theperiod mp
of sequence {pn} and the size L of Table T. Table T is filled
with L elements of sequence {pn} during the Initialization.
After n = LCM(mp, L) iterations of expression (8), where
LCM(mp, L) is the least common multiple of numbers mp
and L, Table T is filled with the same numbers as after the
Initialization. For n > LCM(mp, L), we obtain
U n+LCM(mp,L) = Un. (9)